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A321615 Triangle read by rows: T(n,k) is the number of k X k integer matrices with sum of elements n, with no zero rows or columns, up to row and column permutation.

%I A321615 #22 Jan 17 2024 14:50:15
%S A321615 1,0,1,0,1,1,0,1,2,1,0,1,6,3,1,0,1,9,13,3,1,0,1,17,38,20,3,1,0,1,23,
%T A321615 97,82,23,3,1,0,1,36,217,311,126,24,3,1,0,1,46,453,968,624,151,24,3,1,
%U A321615 0,1,65,868,2825,2637,933,162,24,3,1,0,1,80,1585,7394,10098,4942,1132,165,24,3,1
%N A321615 Triangle read by rows: T(n,k) is the number of k X k integer matrices with sum of elements n, with no zero rows or columns, up to row and column permutation.
%C A321615 Also the number of non-isomorphic multiset partitions of weight n with k parts and k vertices, where the weight of a multiset partition is the sum of sizes of its parts. - _Gus Wiseman_, Nov 18 2018
%H A321615 Andrew Howroyd, <a href="/A321615/b321615.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e A321615 Triangle begins:
%e A321615     1
%e A321615     0  1
%e A321615     0  1    1
%e A321615     0  1    2    1
%e A321615     0  1    6    3    1
%e A321615     0  1    9   13    3    1
%e A321615     0  1   17   38   20    3    1
%e A321615     0  1   23   97   82   23    3    1
%e A321615     0  1   36  217  311  126   24    3    1
%e A321615     0  1   46  453  968  624  151   24    3    1
%e A321615     0  1   65  868 2825 2637  933  162   24    3    1
%t A321615 (* See A318795 for M[m, n, k]. *)
%t A321615 T[n_, k_] := M[k, k, n] - 2 M[k, k-1, n] + M[k-1, k-1, n];
%t A321615 Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 24 2018, from PARI *)
%o A321615 (PARI) \\ See A318795 for M.
%o A321615 T(n, k) = if(k==0, n==0, M(k, k, n) - 2*M(k, k-1, n) + M(k-1, k-1, n));
%o A321615 (PARI) \\ See A340652 for G.
%o A321615 T(n)={[Vecrev(p) | p<-Vec(1 + sum(k=1, n, y^k*(polcoef(G(k, n, n, y), k, y) - polcoef(G(k-1, n, n, y), k, y))))]}
%o A321615 { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 16 2024
%Y A321615 Columns k=0..3 are A000007, A000012, A054974, A054975.
%Y A321615 Row sums are A319616.
%Y A321615 Cf. A007716, A048291, A054976, A057149, A057150, A104601, A120732, A318795, A320808.
%K A321615 nonn,tabl
%O A321615 0,9
%A A321615 _Andrew Howroyd_, Nov 14 2018
%E A321615 Column k=0 inserted by _Andrew Howroyd_, Jan 17 2024